sample.tex

\documentclass{article}
\usepackage{amsmath}
\begin{document}

\def\trace{\operatorname{tr}}

Here is a {\em Macaulay2} example.
<<<R = ZZ[a..i];>>>
We make a generic matrix $m$.
<<<m = matrix{{a,b,c},{d,e,f},{g,h,i}}>>>
The first output line, labelled with ({\tt o2 =}), displays the answer, and
the second output line, labelled with ({\tt o2 :}), displays its {\em class},
or type, and perhaps some auxiliary information.
Now we make the quotient ring of $R$ modulo the $2\times 2$ minors of $m$ and
the equation $\trace m = 1$.
<<<S = R/(minors_2 m + ideal (trace m - 1));>>>
We promote $m$ to $S$, thereby making our generic matrix of rank at most 1 and
of trace 1.
<<<m = promote(m,S)>>>
Now we check whether $m$ is idempotent.
<<<m^2==m>>>
Since the matrix $m$ was generic, we have proven a little theorem, namely, that any $3
\times 3$ matrix with entries in a commutative ring of rank at most 1 and of
trace 1 is idempotent.

<<<exit>>>

Now we've restarted, so {\tt m} should be unset.
<<<m>>>

\end{document}